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Units

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An element of a set is a unit if its multiplicative inverse belongs to that set. (The set of units form a multiplicative group.) Algebraic integer units have complex norm
1
and are thus complex roots of unity.

Units in quadratic integer rings

Imaginary quadratic integer rings have only a few units (see Theorem I1). The primitive root of unity of degree
3
\omega := e^{i \tfrac{2 \pi}{3}} = \frac{-1}{2} + \frac{\sqrt{-3}}{2},

is used in the following table.

Units of imaginary quadratic integer rings
d
Units
≤ @opsp@−@opsp@5 {( − 1)0,( − 1)1} = {1, − 1}
@opsp@−@opsp@3 \{ \omega, - \, \omega, \omega^2, - \, \omega^2, 1, -1 \}
@opsp@−@opsp@2 {( − 1)0,( − 1)1} = {1, − 1}
@opsp@−@opsp@1 {i0,i1,i2,i3} = {1,i, − 1, − i}

Real quadratic integer rings have infinitely many units (see Theorem R1), which are all powers of each other, some multiplied by
1
. For that reason, the table below can't be complete like the table above. An inefficient way to find a unit of a real quadratic integer ring
[2  d ]
other than
1
or
1
is to try each positive value of
b
starting with
1
and going up until
2  1 + db 2
is an integer.

Units of real quadratic integer rings
d
Units
2 1 + \sqrt{2}
3 2 + \sqrt{3}
4 N/A
5 \frac{1}{2} + \frac{\sqrt{5}}{2} (golden ratio)
6 5 + 2 \sqrt{6}
7 8 + 3 \sqrt{7}
8 N/A, but note that 32 − 8(12) = 1
9 N/A
10 3 + \sqrt{10}
11 10 + 3 \sqrt{11}
12 N/A, but 72 − 12(22) = 1
13 \frac{3}{2} + \frac{\sqrt{13}}{2}
14 15 + 4 \sqrt{14}
15 4 + \sqrt{15}
16 N/A
17 4 + \sqrt{17}
18 N/A, but 172 − 18(42) = 1
19 170 + 39 \sqrt{19}
20 N/A, but 92 − 20(22) = 1
21 \frac{5}{2} + \frac{\sqrt{21}}{2}
22 197 + 42 \sqrt{22}
23 24 + 5 \sqrt{23}
24 N/A, but 52 − 24(12) = 1
25 N/A
26 5 + \sqrt{26}
27 N/A, but 262 − 27(52) = 1
28 N/A, but 1272 − 28(242) = 1
29 \frac{5}{2} + \frac{\sqrt{29}}{2}
33 \frac{23}{2} + \frac{4 \sqrt{33}}{2}

Examples

  • The units of
    are
    {1 0} = {1}
    , where
    1 ≡ ei 2π
    . (See natural numbers.)
  • The units of
    are
    {(−1) 0, (−1) 1} = {1, −1}
    , where
    −1 ≡ eiπ
    . (See rational integers.)
  • The units of
    ℤ [ω]
    are
    {ω 0, ω 1, ω 2} = {1, ω, ω 2}
    , where
    ωei
    2π
    3
    . (See Eisenstein integers.)
  • The units of
    ℤ [i]
    are
    {i 0, i 1, i 2, i 3} = {1, i, −1, − i}
    , where
    iei
    π
    2
    . (See Gaussian integers.)
  • The units of
    are
    ℚ \ {0}
    .
  • The units of
    are
    ℝ \ {0}
    .
  • The units of
    are
    ℂ \ {0}
    .

See also

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